Properties of Local Nondeterminism of Gaussian

and Stable Random Fields and Their

Applications

Yimin Xiao ∗

Michigan State University

August 30, 2005

Abstract

In this survey, we first review various forms of local nondeterminismand sectorial local nondeterminism of Gaussian and stable random fields.Then we give sufficient conditions for Gaussian random fields with station-ary increments to be strongly locally nondeterministic (SLND). Finally,we show some applications of SLND in studying sample path propertiesof (N, d)-Gaussian random fields. The class of random fields to which theresults are applicable includes fractional Brownian motion, the Browniansheet, fractional Brownian sheets and so on.

Running head: Local Nondeterminism of Gaussian Random Fields2000 AMS Classification Numbers: Primary 60G15, 60G17, 60G60. Sec-

ondary 60G18, 28A80, 28A78.Key words: Local nondeterminism, Gaussian or stable random field, frac-

tional Brownian motion, the stable sheet, small ball probability, local times,level set, Hausdorff dimension and Hausdorff measure.

1 Introduction

The most important example of self-similar (non-Markovian) Gaussian processesis fractional Brownian motion (fBm) which was first introduced, as a movingaverage Gaussian process, by Mandelbrot and Van Ness (1968)

BH(t) = κH

∫ t

−∞

[((t− s)+)H−1/2 − ((−s)+)H−1/2

]dB(s),

where t+ = max{t, 0}, B is the ordinary Brownian motion and κH > 0 is thenormalizing constant so that E(BH(1)2) = 1, where H ∈ (0, 1) is called the self-similarity index, or Hurst index. Except the case H = 1/2, fBm does not have

∗Research partially supported by NSF grant DMS-0404729.

1

independent increments, it is not a Markov process, nor a semimartingale; seeLin (1995) or Rogers (1997) for a proof of this last fact. Due to its self-similarityand long-range dependence (as H > 1/2), it has been applied to model variousphenomena in telecommunications, turbulence, image processing and finance.As a result, the theory on fractional Brownian motion has been developed sig-nificantly. We refer to Doukhan et al. (2003) for further information.

Moreover, in recent years, many authors have proposed to use more generalself-similar Gaussian processes and random fields as stochastic models in sev-eral different scientific areas; see e.g. Addie et al. (1999), Anh et al. (1999),Benson et al (2004), Bonami and Estrade (2003), Cheridito (2004), Manner-salo and Norros (2002), Mueller and Tribe (2002), just to mention a few. Suchapplications have raised many interesting theoretical questions about Gaussianrandom fields in general.

One of the major difficulties in studying the probabilistic, analytic or statis-tical properties of Gaussian random fields is the complexity of their dependencestructures. As a result, many of the existing tools from theories on Brownianmotion, Markov processes or martingales fail for Gaussian random fields; andone often has to use general principles for Gaussian processes or to develop newtools. In this paper, we show that in many circumstances, the properties of localnondeterminism can help us to overcome this difficulty so that many elegantand deep results of Brownian motion (and Markov processes) can be extendedto Gaussian (or stable) random fields.

The rest of this paper is organized as follows. In Section 2, we recall thedefinitions of various forms of local nondeterminism. In Section 3, we givesufficient conditions for ordinary or strong local nondeterminism to hold forGaussian random fields with stationary increments. In Section 4, we showapplications of the properties of local nondeterminism in studying small ballprobabilities, Hausdorff dimension and exact Hausdorff measure functions ofthe sample paths, and local times of Gaussian random fields.

We end this section with some general notation. Throughout this paper(except in Section 2.4), X = {X(t), t ∈ RN} will denote an (N, d)-Gaussianrandom field, where for every t ∈ RN ,

X(t) =(X1(t), . . . , Xd(t)

), (1.1)

and we will assume E(Xj(t)) ≡ 0 for every 1 ≤ j ≤ d. When N = 1, X is calleda Gaussian process in Rd.

A parameter t ∈ RN is written as t = (t1, . . . , tN ) and if t1 = t2 = · · ·= tN = c ∈ R, then we write t as 〈c〉. There is a natural partial order, “4”, onRN . Namely, s 4 t if and only if s` ≤ t` for all ` = 1, . . . , N . When s 4 t, wedefine the closed interval or rectangle,

[s, t] =N∏

`=1

[s`, t`].

We will let A denote the class of all N -dimensional closed intervals T ⊂ RN .

2

We use 〈·, ·〉 and | · | to denote the ordinary scalar product and the Euclideannorm in Rm respectively, no matter the value of the integer m.

Unspecified positive and finite constants will be denoted by c which may havedifferent values from line to line. Specific constants in Section i will be denotedby c

i,1 , ci,2 , . . .. For two non-negative functions f and g on RN , we denote f ³ gif there exists a finite constant c ≥ 1 such that c−1f(x) ≤ g(x) ≤ c f(x) for allx in some neighborhood of 0.

Acknowledgement. The author thanks Professors Serge Cohen and JacquesIstas for their invitation.

2 Definitions of local nondeterminism

In this section, we recall the definitions of different forms of local nondetermin-ism for Gaussian and stable random fields.

2.1 Local nondeterminism for Gaussian random fields

The concept of local nondeterminism (LND, in short) of a Gaussian process wasfirst introduced by Berman (1973) to unify and extend his methods for study-ing the existence and joint continuity of local times of real-valued Gaussianprocesses. Berman’s definition was later extended by Pitt (1978) and Cuz-ick (1982a) to (N, d)-Gaussian random fields and by Cuzick (1978) to localφ-nondeterminism for an arbitrary positive function φ.

Berman’s definition of LND for Gaussian processes Let X = {X(t), t ∈R+} be a real-valued, separable Gaussian process with mean 0 and let T ⊂ R+

be an open interval. Assume that E[X(t)2] > 0 for all t ∈ T and there existsδ > 0 such that

σ2(s, t) = E[(X(s)−X(t))2

]> 0 for s, t ∈ T with 0 < |s− t| < δ.

Recall from Berman (1973) that X is called locally nondeterministic on T if forevery integer n ≥ 2,

limε→0

inftn−t1≤ε

Vn > 0, (2.1)

where Vn is the relative prediction error:

Vn =Var

(X(tn)−X(tn−1)|X(t1), . . . , X(tn−1)

)

Var(X(tn)−X(tn−1)

) (2.2)

and the infimum in (2.1) is taken over all ordered points t1 < t2 < · · · < tn inT with tn − t1 ≤ ε. Roughly speaking, (2.1) means that a small increment ofthe process X is not almost relatively predictable based on a finite number ofobservations from the immediate past.

3

It follows from Berman (1973, Lemma 2.3) that (2.1) is equivalent to thefollowing property which says that X has locally approximately independentincrements: for any positive integer n ≥ 2, there exist positive constants cn andδ (both may depend on n) such that

Var( n∑

j=1

uj

(X(tj)−X(tj−1)

)) ≥ cn

n∑

j=1

u2j σ2(tj−1, tj) (2.3)

for all ordered points 0 = t0 < t1 < t2 < · · · < tn in T with tn − t1 < δ and alluj ∈ R (1 ≤ j ≤ n). We refer to Nolan (1989, Theorem 2.6) for a proof of theabove equivalence in much more general setting.

Local nondeterminism for Gaussian random fields In order to study thejoint continuity of the local times of an (N, d)-Gaussian random field X ={X(t), t ∈ RN}, Pitt (1978) extended Berman’s definition (2.1) of LND to therandom field setting; see also Geman and Horowitz (1980). Assume that T ∈ Ais an interval and for all s 6= t ∈ T , the covariance matrix of X(s) − X(t) ispositive definite and is denoted by Σ2(s, t). Then there is a non-singular matrixΣ(s, t) such that Σ(s, t)Σ′(s, t) = Σ2(s, t).

According to Pitt (1978), a Gaussian random field X as above is called locallynondeterministic on T if for every integer n ≥ 2, there exist positive constantscn and δn such that for all u = (u1, . . . , un) ∈ Rnd\{0},

Var( n∑

j=1

〈uj , Σ−1j (X(tj)−X(tj−1))〉

)≥ cn

n∑

j=1

|uj |2 (2.4)

whenever the points t1, t2, . . . , tn are distinct and all lie in a sub-interval of Twith side-length at most δn, and satisfy

|tj − tj−1| ≤ |tj − ti| for all 1 ≤ i < j ≤ n. (2.5)

Note that (2.5) introduces a partial order among t1, . . . , tn ∈ RN ; and there areat least n different ways to order them using (2.5).

Cuzick (1982a) gives another definition of local nondeterminism: an (N, d)-Gaussian random field X is locally nondeterministic on T if for all integersn > 1, there exist cn > 0 and δn > 0 (depending only on n) such that for anyt1, . . . , tn ∈ T with |tj − tn| ≤ δn, the conditional vector X(tn) given X(tj),j = 1, . . . , n− 1 satisfies

detCov(X(tn)

∣∣X(tj), 1 ≤ j ≤ n− 1) ≥ cn detCov

(X(tn)−X(t∗)

), (2.6)

where t∗ = ti if |ti−tn| = infj<n |tj−tn| and detCov(Z) denote the determinantof the covariance matrix of the random vector Z.

Note that when d = 1 or, d > 1 and X has independent components, Theo-rem 2.6 of Nolan (1989) implies that (2.4) and (2.6) are equivalent. In general,however, it does not seem clear how these two definitions are related.

4

Remark 2.1 Both definitions of Pitt and Cuzick are applicable to all (N, d)-Gaussian random fields. Even though so far most authors have been workingonly with (N, d)-Gaussian random fields with independent components, it hasbecome clear that one also needs to study (N, d)-Gaussian fields with dependentcomponents. An interesting example of such Gaussian random fields is the op-erator fractional Brownian motion defined in Mason and Xiao (2002). It wouldbe interesting to know whether it is LND in the sense of Pitt and/or Cuzick.An affirmative answer will be useful to establish many interesting sample pathproperties of operator fractional Brownian motion. ¤

The inequalities (2.3) and (2.4) have played significant roles in the worksof Berman (1969–1973) and Pitt (1978) on local time theory of a large classof Gaussian random fields. Their results, in turn, imply irregularity and frac-tal properties of the sample paths of Gaussian random fields. See Geman andHorowitz (1980), Adler (1981), Geman et al. (1984) and the references thereinfor further information. Moreover, local nondeterminism has been applied byRosen (1984) and Berman (1991) to study the existence and regularity of self-intersection local times, by Kahane (1985) to study the image and level setsof fractional Brownian motion, and by Monrad and Pitt (1987) to prove uni-form Hausdorff dimension results for the image and inverse image of Gaussianrandom fields. Because of its various applications, it has been an interestingquestion to determine when a given Gaussian process is locally nondeterminis-tic. Some sufficient conditions for real-valued Gaussian processes to be locallynondeterministic can be found in Berman (1973, 1988, 1991), Cuzick (1978),Pitt (1978).

k-th order local nondeterminism Berman’s definition of LND was extendedby Cuzick (1978) who defined local φ-nondeterminism for real-valued Gaussianprocesses by replacing the variance function σ2(tn, tn−1) in (2.2) by φ(tn−tn−1),where φ is an arbitrary positive function. Furthermore, he has defined the so-called kth order local φ-nondeterminism, not for the process X itself, but forthe k-th divided differences of X; see Cuzick (1978, p.73) for details. He hasgiven sufficient conditions for a stationary Gaussian processes to have this kthorder LND property and then applied it to estimate the moments of the numberN(0, T ) of zero crossings of a smooth stationary Gaussian process X in timeinterval [0, T ]. In particular, he has provided verifiable sufficient conditionsfor the finiteness of the k-th factorial moment Mk(0, T ) of N(0, T ); see alsoCuzick (1975) and Miroshin (1977). Even though the rest of this paper will notdiscuss the k-th order local φ-nondeterminism any further, we mention that, inorder to study the rate of growth of Mk(0, T ) as a function of k, Cuzick (1978,p. 81) has noticed that the k-th order local φ-nondeterminism is not enoughand has suggested to use a notion of k-th order strong φ-local nondeterminism.See Cuzick (1977) for some partial results along this direction on a stationaryGaussian process X such that X ′ exists in the quadratic mean sense. It wouldbe interesting to study this problem under the more general setting of Section2.2.

5

2.2 Strong φ-local nondeterminism for Gaussian randomfields

There are some drawbacks in the definitions of local nondeterminism in Sec-tion 2.1: one is that the liminf in (2.1) and the constant cn in (2.3) depend onthe number of “time” points; the other is that there are many different waysto order n points in RN using (2.5). Because of these, the properties of lo-cal nondeterminism defined by Berman (1973), Pitt (1978) and Cuzick (1978,1982a) are not enough for establishing fine regularity properties such as the lawof the iterated logarithm and the modulus of continuity for the local times orself-intersection local times of Gaussian random fields. For studying these andmany other problems on Gaussian random fields, the concept of strong local non-determinism (SLND) has proven to be more appropriate. See Cuzick (1982b),Monrad and Pitt (1987), Csorgo et al. (1995), Monrad and Rootzen (1995),Talagrand (1995, 1998), Xiao (1996, 1997a, b, c), Kasahara et al. (1999), Xiaoand Zhang (2002), just to mention a few. In Section 4, we will address some ofthese aspects.

The following definition of the strong local φ-nondeterminism (SLφND) wasessentially given by Cuzick and DuPreez (1982) for Gaussian processes (i.e.,N = 1). For Gaussian random fields, Definition 2.2 is more general than thedefinition of strong local α-nondeterministism of Monrad and Pitt (1987).

Definition 2.2 Let X = {X(t), t ∈ RN} be a real-valued Gaussian randomfield with 0 < E[X(t)2] < ∞ for all t ∈ T , where T ∈ A is an interval. Let φbe a given function such that φ(0) = 0 and φ(r) > 0 for r > 0. Then X is saidto be strongly locally φ-nondeterministic (SLφND) on T if there exist positiveconstants c2,1 and r0 such that for all t ∈ T and all 0 < r ≤ min{|t|, r0},

Var(X(t)|X(s) : s ∈ T, r ≤ |s− t| ≤ r0

) ≥ c2,1 φ(r). (2.7)

Remark 2.3 By modifying the proof of Proposition 7.2 of Pitt (1978), we canverify that if (2.7) holds and T is bounded away from 0, then for all n ≥ 2 thereexists a constant c2,2 = c2,2(n) > 0 such that

Var( n∑

j=1

uj(X(tj)−X(tj−1)))≥ c2,2

n∑

j=1

u2j φ(|tj − tj−1|) (2.8)

for all uj ∈ R and tj ∈ T (j = 1, . . . , n) satisfying (2.5). That is, X is locallyφ-nondeterministic on T in the sense of Section 2.1. On the other hand, Cuzick(1977) has given an example of stationary Gaussian process X = {X(t), t ∈ R}in R that satisfies (2.8) for each fixed integer n and a function φ ³ σ2, while theconditional variance in the left-hand side of (2.7) equals 0. Hence SLND (2.7)is strictly stronger than Berman’s LND (2.1) or (2.3). ¤

6

Remark 2.4 When N = 1, one could also define X to be strongly locallyφ-nondeterministic when the constant cn in (2.3) (with σ2 replaced by φ) is in-dependent of n. Clearly, this condition implies (2.7). It is not known whether theconverse is true; see Remark 2.3 for a weaker result. Even though this alterna-tive way of defining SφLND is not needed for Gaussian processes, a modificationof this is useful for stable processes; see Section 2.4. ¤

Remark 2.5 We mention that in the studies of Gaussian processes X ={X(t), t ∈ R}, due to the simple order structure of R, it is sometimes enoughto assume that X is one-sided strongly locally φ-nondeterministic, namely, forsome constant c2,3 > 0

Var(X(t)|X(s) : s ∈ T, r ≤ t− s ≤ r0

) ≥ c2,3 φ(r); (2.9)

see Cuzick (1978), Berman (1972, 1978), Monrad and Rootzen (1995). WhenX = {X(t), t ∈ R} is a Gaussian process with stationary increments, some suffi-cient conditions in terms of the variance function σ2(h) = E

[(X(t+h)−X(t)

)2]for the one-sided strong local nondeterminism have been obtained earlier. Mar-cus (1968a) and Berman (1978) have proved that if σ(h) → 0 as h → 0 andσ2(h) is concave on (0, δ) for some δ > 0, then X is one-sided strongly locallyφ-nondeterministic for φ(r) = σ2(r). ¤

The most important example of SLND Gaussian random field is the N -parameter fractional Brownian motion BH = {BH(t), t ∈ RN} of index H(0 < H < 1). This is a centered, real-valued Gaussian random field withcovariance function

E(BH(t)BH(s)

)=

12(|t|2H + |s|2H − |t− s|2H

).

The strong local φ-nondeterminism of BH with φ(r) = r2H follows from Lemma7.1 of Pitt (1978), where the self-similarity of BH has played an essentialrole. For a stationary Gaussian process X = {X(t), t ∈ R}, Cuzick andDuPreez (1982) have given a sufficient condition for X to be strongly locallyφ-nondeterministic in terms of its spectral measure F . More precisely, they haveproved that if the absolutely continuous part of dF (λ) has the property that

dF (λ/r)φ(r)

≥ h(λ)dλ ∀ 0 < r ≤ r0 (2.10)

and ∫ ∞

0

log h(λ)1 + λ2

dλ > −∞, (2.11)

then X is SLφND. Their proof uses the ideas from Cuzick (1977) and relies onthe special properties of stationary Gaussian processes. Note that when N = 1,the strong local r2H -nondeterminism of BH can also be derived from the aboveresult of Cuzick and DuPreez (1982) by using the Lamperti transformation. Thisapproach can be applied to study self-similar Gaussian processes in general.

7

In Section 3 we will give a sufficient condition for Gaussian random fieldswith stationary increments to be strongly locally nondeterministic.

2.3 Sectorial local nondeterminism for anisotropic Gauss-ian random fields

In Definition 2.2, (2.7) measures the prediction error in terms of the distancebetween t and the region where the information is known. This works if theGaussian random field X has certain approximately isotropic property, but cannot be expected to hold for general anisotropic random fields. In fact, it hasbeen well-known that the Brownian sheet does not have this type of stronglocal nondeterminism. This accounts for the significant difference between theexisting methods for studying the fractional Brownian motion and the Browniansheet.

Recently, Khoshnevisan and Xiao (2004b) have shown that the Browniansheet possesses the so-called sectorial local-nondeterminism. This property leadsto a unification of many of the methods developed for fractional Brownian mo-tion and those for the Brownian sheet and to solutions of several problems onthe image and multiple points of the Brownian sheet. See Khoshnevisan andXiao (2004b), Khoshnevisan, Wu and Xiao (2005) for further information.

In the following, we will discuss sectorial local nondeterminism for fractionalBrownian sheets. Recall that, for a given vector ~H = (H1, . . . , HN ) ∈ (0, 1)N , areal-valued fractional Brownian sheet B

~H0 = {B ~H

0 (t), t ∈ RN+} with Hurst index

~H is a centered Gaussian random field with covariance function given by

E[B

~H0 (s)B ~H

0 (t)]

=N∏

`=1

12

(s2H`

` + t2H`

` − |s` − t`|2H`

), s, t ∈ RN

+ . (2.12)

It follows from (2.12) that B~H0 (t) = 0 a.s. for every t ∈ ∂RN

+ , where ∂RN+

denotes the boundary of RN+ .

Let B~H1 , . . . , B

~Hd be d independent copies of B

~H0 . Then the Gaussian random

field B~H = {B ~H(t), t ∈ RN

+} with values in Rd defined by

B~H(t) = (B ~H

1 (t), . . . , B ~Hd (t)), ∀ t ∈ RN

+ (2.13)

is called an (N, d)-fractional Brownian sheet with Hurst index ~H = (H1, . . . ,

HN ). It follows from (2.12) that B~H has the following operator-self-similarity:

for any N × N diagonal matrix A = (aij) with aii = ai > 0 for all 1 ≤ i ≤ Nand aij = 0 if i 6= j, we have

{B

~H(At), t ∈ RN} d=

{ N∏

j=1

aHj

j B~H(t), t ∈ RN

}, (2.14)

where Xd= Y means that the two processes have the same finite dimensional

distributions. Moreover, for every ` = 1, . . . , N , B~H is a fractional Brownian

motion in Rd of Hurst index H` along the direction of the `th axis.

8

If N > 1 and H1 = · · · = HN = 1/2, then BH is the (N, d)-Brownian sheet.See Orey and Pruitt (1973) and Khoshnevisan (2002) for systematic accountson the Brownian sheets.

Fractional Brownian sheets arise naturally in many areas such as in stochas-tic partial differential equations [cf. Øksendal and Zhang (2000), Hu, Øksendaland Zhang (2000)] and in the studies of most visited sites of symmetric Markovprocesses [cf. Eisenbaum and Khoshnevisan (2002)]. One of the important fea-tures of B

~H is that, when H1, . . . , HN are different, it has different probabilisticand analytic behaviors along different directions and thus is highly anisotropic.Recently, there have been interest in using anisotropic Gaussian random fieldsto model bone structure [Bonami and Estrade (2003)] and aquifer structure inhydrology [Benson et al. (2004)]. We believe that the results and techniquesfor characterizing the anisotropic properties of the fractional Brownian sheet interms of ~H will also be helpful for studying other types of anisotropic Gaussianrandom fields.

The main tools for analyzing the dependence structure of B~H0 are the fol-

lowing stochastic integral representations. They can be proved by verifying thecovariance functions.

• Moving average representation

B~H0 (t) = c−1

2,4

∫ t1

−∞· · ·

∫ tN

−∞

N∏

`=1

gH`

(t`, s`)W (ds), (2.15)

where W = {W (s), s ∈ RN} is a standard real-valued Brownian sheet and forH ∈ (0, 1) and s, t ∈ R,

gH (t, s) =((t− s)+

)H−1/2 − ((−s)+

)H−1/2,

with s+ = max{s, 0}, and where c2,4 is the normalizing constant given by

c22,4

=∫ 1

−∞· · ·

∫ 1

−∞

[ N∏

`=1

gH`

(1, s`)]2

ds.

• Harmonizable representation

B~H0 (t) = c−1

2,5

∫

RN

N∏

j=1

eitjλj − 1|λj |Hj+

12

W (dλ), (2.16)

where W is the Fourier transform of white noise in RN and c2,5 > 0 is thenormalizing constant so that Var

(B

~H0 (〈1〉)) = 1. This representation for B

~H0 is

proved by Herbin (2004).The following sectorial LND of fractional Brownian sheet is proved by Wu

and Xiao (2005), extending a results of Khoshnevisan and Xiao (2004b) on theBrownian sheet.

9

Lemma 2.6 Let B~H0 = {B ~H

0 (t), t ∈ RN+} be a fractional Brownian sheet in R

with Hurst index ~H = (H1, . . . ,HN ) ∈ (0, 1)N . Then for any ε > 0, there is aconstant c2,6 > 0 such that for all integers n ≥ 2, t1, . . . , tn ∈ [ε,∞)N ,

Var(B

~H0 (tn)

∣∣∣B ~H0 (tj), 1 ≤ j ≤ n− 1

)≥ c2,6

N∑

`=1

min0≤j≤n−1

|tn` − tj` |2H` , (2.17)

where t0` = 0 for every ` = 1, . . . , N .

The proof of Lemma 2.6 makes use of the harmonizable representation ofB

~H0 and a Fourier analytic argument. This lemma plays key roles in Ayache,

Wu and Xiao (2005) who verify a conjecture of Xiao and Zhang (2002) on thejoint continuity of local times of a fractional Brownian sheet B

~H , and in Wuand Xiao (2005) who study the geometric properties of the sample paths of B

~H .

2.4 Local nondeterminism for stable processes

In this subsection, we will discuss briefly the properties of local nondeterminismfor stable random fields. First we mention the following papers which are closelyrelated to the topics of this paper, but will not be further addressed becauseall the random fields considered there possess certain Markovian nature. Ehm(1981) has established many deep results on the sample path properties of thestable sheet and his arguments rely crucially on the property of independentincrements of the stable sheet. Khoshnevisan, Xiao and Zhong (2003a, b) haveextended several of Ehm’s results to additive Levy processes and have also es-tablished some useful connections between hitting probabilities and a class ofnatural capacities. Mountford and Nualart (2004) and Mountford (2004) de-termine the exact Hausdorff measure functions for the level sets of an additiveBrownian motion and additive stable processes, respectively. The property ofindependent increments of Levy processes and a certain type of Markov prop-erty have played crucial roles in the work of these authors. We refer to thesurvey papers of Khoshnevisan and Xiao (2004a) and Xiao (2004) for furtherinformation along this line.

The class of symmetric α-stable (SαS) self-similar processes and randomfields is very large; see Samorodnitsky and Taqqu (1994) for a systematic ac-count. Of special interest are the linear fractional stable motion and harmoniz-able fractional stable motion introduced by Taqqu and Wolpert (1983), Maejima(1983), Cambanis and Maejima (1989), respectively. They are natural stableanalogues of fractional Brownian motion.

Compared to Gaussian random fields, much less about the probabilistic,analytic and statistical properties of such stable random fields has been known.We believe that an appropriate notion of strong local nondeterminism for stablerandom fields will be helpful to solve several open problems on local timesand self-intersection local times, as well as to investigate other sample pathproperties.

10

The notion of local nondeterminism has been extended to SαS processes andrandom fields by Nolan (1988, 1989), and has proven to be a useful tool instudying the local times and self-intersection local times of certain self-similarstable processes with stationary increments. See, for example, Kono and Shieh(1993), Shieh (1993) and Xiao (1995).

One of the difficulties of extending LND from Gaussian random fields toα-stable random fields X = {X(t), t ∈ RN} is that, when 0 < α < 2, thereis no covariance to measure dependence of X(t1), . . . , X(tn). Nolan (1989) hasrelied on the Lα-representations of symmetric α-stable random fields [see Hardin(1982) or Samorodnitsky and Taqqu (1994)] and the approximation propertiesof normed or quasi-normed linear spaces.

We first consider the case N = 1. Let T ⊂ R be a closed interval. Thefollowing definition is due to Nolan (1989, Definition 3.1) which reduces to (2.3)when α = 2.

Definition 2.7 A real-valued SαS process X = {X(t), t ∈ R} is called locallynondeterministic on T if for every integer n > 1, there exists a constant cn ≥ 1depending on n only such that for all sufficiently close t1 < t2 < . . . < tn in T ,

∣∣∣E(eicn u1X(t1)

) n∏

j=2

E(eicn uj(X(tj)−X(tj−1))

)∣∣∣

≤∣∣∣E exp

{i(u1X(t1) +

n∑

j=2

uj(X(tj)−X(tj−1)))}∣∣∣

≤∣∣∣E

(eic−1

n u1X(t1)) n∏

j=2

E(eic−1

n uj(X(tj)−X(tj−1)))∣∣∣

(2.18)

for all uj ∈ R (j = 1, . . . , n).

Hardin (1982) proved that for every real-valued, separable in probability,SαS process X = {X(t), t ∈ R}, there exist a measure space (E,B, µ) and acollection of real-valued functions {κ(t, ·), t ∈ R} ⊆ Lα(E,B, µ) such that for allintegers n ≥ 1 the joint distribution of X(t1), . . . , X(tn) is determined by

E exp(i

n∑

j=1

ujX(tj))

= exp(−

∥∥∥n∑

j=1

ujκ(tj)∥∥∥

α

α

), (2.19)

where ‖ · ‖α is the quasi-norm in Lα(E,B, µ) and κ(tj)=κ(tj , ·). Based onthis fact, Nolan (1989) proves that (2.18) in Definition 2.7 is equivalent to thefollowing: for every integer n > 1, there exists a constant c2,7 = c2,7(n) ≥ 1

11

depending on n only such that

c−12,7

(∥∥u1κ(t1)∥∥

α+

n∑

j=2

∥∥uj(κ(tj)− κ(tj−1))∥∥

α

)

≤∥∥∥u1κ(t1) +

n∑

j=2

uj

(κ(tj)− κ(tj−1)

)∥∥∥α

≤ c2,7

(∥∥u1κ(t1)∥∥

α+

n∑

j=2

∥∥uj(κ(tj)− κ(tj−1))∥∥

α

)

(2.20)

for all uj ∈ R and all t1 < t2 < . . . < tn in T such that tn − t1 is sufficientlysmall. Nolan (1989, Theorem 3.2) also gives some other equivalent definitionsof LND for real-valued SαS processes.

An (N, d)-random field X = {X(t), t ∈ RN} is called an (N, d, α)-stablefield if for all integers n ≥ 1, t1, . . . , tn ∈ RN and u1, . . . , un ∈ Rd, the ran-dom variables

∑nj=1 〈uj , X(tj)〉 are SαS random variables. For a given measure

space (E,B, µ), let Lα(E,B, µ;Rd) denote the collections of Rd-valued func-tions κ(·) such that κ(·) = (κ1(·), . . . , κd(·)) and κj(·) ∈ Lα(E,B, µ) for everyj = 1, . . . , d. It is possible to represent an (N, d, α)-stable field X in someLα(E,B, µ;Rd). That is, there is a family of functions {κ(t, ·), t ∈ RN} in thespace Lα(E,B, µ;Rd) such that

E exp(i

n∑

j=1

〈uj , X(tj)〉)

= exp(−

∥∥∥n∑

j=1

〈uj , κ(tj)〉∥∥∥

α

α

). (2.21)

Nolan (1989, Definition 3.3) defines LND of an (N, d, α)-stable field X in termsof the family {κ(t, ·), t ∈ RN}.Definition 2.8 An (N, d, α)-stable field X is called locally nondeterministic onan interval T ∈ A if its representation {κ(t, ·), t ∈ RN} satisfies the followingconditions:

(a) ‖κj(t)‖α > 0 for all t ∈ T and j = 1, . . . , d.

(b) ‖κj(s) − κj(t)‖α > 0 for all s, t ∈ T with |s − t| sufficiently small andj = 1, . . . , d.

(c) For all integers n ≥ 1, arbitrary t1, . . . , tn ∈ T and all j = 1, . . . , d, defineMn

j to be the subspace of Lα(E,B, µ) spanned by {κl(tk) : 1 ≤ l ≤ d, 1 ≤k ≤ n and (l, k) 6= (j, n)}. Then for all j = 1, . . . , d,

inft1∈T

‖κj(t1)−M1j ‖α

‖κj(t1)‖α> 0 (2.22)

and

lim inf‖κj(tn)−Mn

j ‖α

‖κj(tn)− κj(tn−1)‖α> 0, (2.23)

12

where the liminf is taken over all t1, . . . , tn ∈ T satisfying (2.5) with |tn−t1| → 0, and ‖κj(tn) − Mn

j ‖α denotes the “Lα-distance” between κj(tn)and Mn

j .

Nolan (1989) shows that Condition (c) in Definition 2.8 is equivalent to theassumption that X has, in certain sense, approximately independent compo-nents and approximately independent increments. This is useful for establish-ing the joint continuity of local times of several classes of stable processes orstable random fields; see Nolan (1989), Kono and Shieh (1993), Shieh (1993)and Xiao (1995). However, as in the Gaussian case, this LND property is notuseful for obtaining sharp uniform and/or local growth properties of the localtimes or self-intersection local times of SαS processes and (N, d, α)-stable fields;see Dozzi and Soltani (1999, section 4) for related remarks. One needs to havea notion of strong local nondeterminism.

When N = 1, we recall Remark 2.4 and may define conveniently that an SαSprocess X is strongly locally nondeterministic on T if there exists a constantc2,8 > 0 such that the following hold: for all integers n ≥ 2, we can find anonsingular n× n matrix A such that for all t1 < t2 < . . . < tn in T sufficientlyclose and all uj ∈ R (j = 1, . . . , n),

∣∣∣E exp{

i(u1X(t1) +

n∑

j=2

uj(X(tj)−X(tj−1)))}∣∣∣

≤∣∣∣E

(eic2,8 v1X(t1)

) n∏

j=2

E(eic2,8 vj(X(tj)−X(tj−1))

)∣∣∣,(2.24)

where (v1, . . . , vn) = (u1, . . . , un)A.Dozzi and Soltani (1999) have studied a class of moving average (MA) SαS

processes X of the form X = X1 + X2, where X1 and X2 are two independentMA-stable processes, X1 is strongly locally nondeterministic in the above senseand X2 is arbitrary. They showed that the arguments of Berman (1973) andEhm (1981) can be modified to prove uniform and local Holder conditions forthe local times of X.

In light of the theory on Gaussian random fields, there should be severaldifferent senses of strong local nondeterminism for (N, d, α)-stable fields. Forsimplicity, we start by considering only isotropic (N, 1, α)-stable fields with sta-tionary increments. It seems natural to define the strong local φ-nondeterminismfor such SαS random fields as follows.

Definition 2.9 Let X = {X(t), t ∈ RN} be an (N, 1, α)-stable field with sta-tionary increments and X(0) = 0. Let φ : R+ → R+ be a given function suchthat φ(0) = 0 and φ(r) > 0 for r > 0 and let T ∈ A. Then X is said to bestrongly locally φ-nondeterministic (SLφND) on T if, in addition to (a) and(b) in Definition 2.8, there exists a constant c2,9 > 0 such that for all integersn ≥ 1, all t, s1, . . . , sn ∈ T sufficiently close,

‖κ(t)−Mn‖αα ≥ c2,9 φ

(min

0≤j≤n|t− sj |), (2.25)

13

where Mn denotes the subspace of Lα(E,B, µ) spanned by {κ(s1), . . . , κ(sn)}and s0 = 0.

It would be very interesting to determine which classes of self-similar stablerandom fields are strongly locally nondeterministic in the above sense and to ex-plore various applications of SLND in the studies of stable random fields. We be-lieve that the linear fractional stable motions (or fields) and harmonizable frac-tional stable motions (or fields) with Hurst index H ∈ (0, 1) [cf. Samorodnitskyand Taqqu (1994), Kokoszka and Taqqu (1994) and Nolan (1989)] are stronglylocally φ-nondeterministic with φ(r) = rH . So far, I have only been able toprove this for harmonizable fractional (N, 1, α)-stable fields when α ∈ [1, 2] anda one-sided SLND for (1, 1, α)-linear fractional stable motion when α ∈ (0, 2].

On the other hand, the (N, 1, α)-stable sheet Zα = {Z(t), t ∈ RN+} defined

in Ehm (1981), which contains the Brownian sheet as a special case, is notstrongly locally φ-nondeterministic in the sense of Definition 2.9. Moreover,similar to (2.15) and (2.16), we can define two classes of (real-valued) anisotropicfractional stable sheets using stochastic integration with respect to an (N, 1, α)-stable sheet Zα or a complex-valued SαS random measure Zα. They are naturalextensions of fractional Brownian sheets to stable random fields.

• Moving average fractional stable sheets: for any given 0 < α < 2 and~H = (H1, . . . , HN ) ∈ (0, 1)N , we define a stable random field Z

~H = {Z ~H(t), t ∈RN

+} with values in R by

Z~H(t) =

∫ t1

−∞· · ·

∫ tN

−∞

N∏

`=1

hH`(t`, s`)Zα(ds), (2.26)

where Zα = {Zα(s), s ∈ RN} is a symmetric (N, 1, α)-stable sheet and forH ∈ (0, 1) and s, t ∈ R,

hH (t, s) = a{(

(t− s)+)H−1/α − (

(−s)+)H−1/α

}

+ b{(

(t− s)−)H−1/α − (

(−s)−)H−1/α

},

(2.27)

where a, b ∈ R are constants and t− = max{−t, 0}. Using (2.26) and the self-similarity of Zα, we can verify that the (N, 1, α)-stable field Z

~H is operatorself-similar in the sense of (2.14), and along each direction of RN

+ , Z~H becomes

a real-valued linear fractional stable motion. We will call Z~H = {Z ~H(t), t ∈ RN

+}an (N, 1, α)-moving average fractional stable sheet.

• Harmonizable fractional stable sheets: for any given 0 < α < 2 and~H = (H1, . . . ,HN ) ∈ (0, 1)N , we define the harmonizable fractional stable sheetZ

~H = {Z ~H(t), t ∈ RN+} with values in R by

Z~H(t) = Re

∫

RN

N∏

j=1

eitjλj − 1|λj |Hj+

1α

Zα(dλ), (2.28)

14

where Zα is a complex-valued SαS random measure. We refer to Samorodnitskyand Taqqu (1994, Chapter 6) for definition and basic properties of complex-valued SαS random measure and the corresponding stochastic integrals.

Similar to the moving average fractional stable sheet, we can verify that Z~H

is operator self-similar in the sense of (2.14). Along each direction of RN+ , Z

~H

becomes a real-valued harmonizable fractional stable motion.Note that, unlike the Gaussian case where both (2.15) and (2.16) determine

(up to a constant) the same fractional Brownian sheet, the moving average andharmonizable fractional stable sheets with the same α ∈ (0, 2) and Hurst index~H are different random fields. This is true even for N = 1; see Samorodnitskyand Taqqu (1994, page 358).

Based on the studies of fractional Brownian sheets, we believe that an ap-propriate definition of sectorial local nondeterminism should be introduced andit will be useful for studying various sample path properties of such anisotropicstable random fields. This problem will be studied elsewhere, and the rest ofthe paper deals with Gaussian random fields only.

3 Spectral conditions for strong local nondeter-minism of Gaussian random fields

As pointed out by Cuzick and DuPreez (1982, p. 811), it appears to be diffi-cult to establish conditions under which general Gaussian processes possess thevarious forms of strong local nondeterminism. In this section we provide suffi-cient conditions for a real-valued Gaussian random field X = {X(t), t ∈ RN}with stationary increments to be strongly locally φ-nondeterministic. In par-ticular, we show that a spectral condition similar to that of Berman (1988)for ordinary LND of Gaussian processes actually implies that X is strongly lo-cally φ-nondeterministic and, importantly, φ(r) is comparable to the variancefunction σ2(h) with |h| = r.

Similar methods, combined with the arguments in Wu and Xiao (2005), canbe modified to study the sectorial local nondeterminism of anisotropic Gaussianrandom fields with stationary increments or Gaussian random fields of fractionalBrownian sheet type. By the latter, I mean their covariance functions are definedas tensor products of covariance functions of Gaussian processes with stationaryincrements. There are many interesting questions on such anisotropic Gaussianrandom fields due to their various applications; see Bonami and Estrade (2003),Cheridito (2004), Mannersalo and Norros (2002), Mueller and Tribe (2002) andthe references therein.

Let X = {X(t), t ∈ RN} be a real-valued, centered Gaussian random fieldwith X(0) = 0. We assume that X has stationary increments and continuouscovariance function R(s, t) = E

[X(s)X(t)

]. According to Yaglom (1957), R(s, t)

can be represented as

R(s, t) =∫

RN

(ei〈s,λ〉 − 1)(e−i〈t,λ〉 − 1)∆(dλ) + 〈s,Qt〉, (3.1)

15

where Q is an N ×N non-negative definite matrix and ∆(dλ) is a nonnegativesymmetric measure on RN\{0} satisfying

∫

RN

|λ|21 + |λ|2 ∆(dλ) < ∞. (3.2)

The measure ∆ is called the spectral measure of X.It follows from (3.1) that X has the following stochastic integral representa-

tion:X(t) =

∫

RN

(ei〈t,λ〉 − 1)W (dλ) + 〈Y, t〉, (3.3)

where Y is an N -dimensional Gaussian random vector with mean 0 and W (dλ)is a centered complex-valued Gaussian random measure which is independentof Y and satisfies

E(W (A)W (B)

)= ∆(A ∩B) and W (−A) = W (A)

for all Borel sets A, B ⊆ RN . From now on, we will assume Y = 0. Conse-quently, we have

σ2(h) = E[(

X(t + h)−X(t))2] = 2

∫

RN

(1− cos 〈h, λ〉) ∆(dλ). (3.4)

If the function σ2(h) only depends on |h|, then X is called an isotropic randomfield. It is important to note that σ2(h) is a negative definite function andcan be viewed as the characteristic exponent of a symmetric infinitely divisibledistribution; see Berg and Forst (1975) for more information on negative definitefunctions.

The main results of this section are Theorems 3.1 and 3.4. They give verifi-able conditions for a Gaussian random field to be strongly locally nondetermin-istic in terms of its spectral measure.

Theorem 3.1 Let X = {X(t), t ∈ RN} be a mean zero, real-valued Gaussianrandom field with stationary increments and X(0) = 0, and let f be the densityfunction of the absolutely continuous part of the spectral measure ∆ of X. As-sume that there exist two non-decreasing functions φ(r) and q(r) : R+ → R+

satisfying the following conditions:

f(λ/r)φ(r)

≥ rN

q(|λ|) , ∀ r ∈ (0, 1] and λ ∈ RN (3.5)

and there exists a positive and finite constant η such that

q(r) ≤ rη for all r > 0 large enough. (3.6)

Then for every interval T ∈ A, there exists a constant 0 < c3,1 < ∞ such thatfor all t ∈ T\{0} and all 0 < r ≤ min{1, |t|},

Var(X(t)|X(s) : s ∈ T, |s− t| ≥ r

) ≥ c3,1 φ(r). (3.7)

In particular, X is strongly locally φ-nondeterministic on T .

16

To prove Theorem 3.1, we note that, in the Hilbert space setting, the con-ditional variance in (3.7) is the square of the L2(P)-distance of X(t) from thesubspace generated by {X(s) : s ∈ T, |s− t| ≥ r}. Hence it is sufficient to showthat there exists a positive constant c3,2 such that for all integers n ≥ 1, ak ∈ Rand sk ∈ T satisfying |sk − t| ≥ r, (k = 1, 2, . . . , n),

E(X(t)−

n∑

k=1

akX(sk))2

≥ c3,2 φ(r). (3.8)

It follows from (3.1) or (3.3) that

E(X(t)−

n∑

k=1

akX(sk))2

=∫

RN

∣∣∣ei〈t,λ〉 − 1−n∑

k=1

ak

(ei〈sk,λ〉 − 1

)∣∣∣2

∆(dλ)

≥∫

RN

∣∣∣ei〈t,λ〉 −n∑

k=0

ak ei〈sk,λ〉∣∣∣2

f(λ)dλ, (3.9)

where a0 = −1 +∑n

k=1 ak and s0 = 0. This part of the proof goes backto Kahane (1985). The last integral can be estimated using the ideas in Pitt(1975, 1978) and Kahane (1985); see Xiao (2005) for a complete proof.

In order to apply Theorem 3.1 to investigate the sample path properties ofthe Gaussian random field X, we need to study the relationship between φ(|h|)and the function σ2(h). In the following, we assume that the spectral measure∆ is absolutely continuous and its density function f(λ) satisfies the followingcondition [when N = 1, this is due to Berman (1988)]:

0 < α =12

lim infλ→∞

βN |λ|Nf(λ)∆{ξ : |ξ| ≥ |λ|} ≤

12

lim supλ→∞

βN |λ|Nf(λ)∆{ξ : |ξ| ≥ |λ|} = α < 1,

(3.10)where β1 = 2 and for N ≥ 2, βN = µ(SN−1) is the surface area [i.e., the(N −1)-dimensional Lebesgue measure) of SN−1. At the end of this section, wewill give some examples of Gaussian random fields satisfying (3.10).

In the rest of this section, we define φ(r) = ∆{ξ : |ξ| ≥ r−1} and φ(0) =0. Then the function φ is non-decreasing and left continuous on [0,∞). Thefollowing lemma lists some useful properties of φ.

Lemma 3.2 Assume the condition (3.10) holds. Then for every 0 < ε <2min{α, 1−α}), there exists a constant r0 > 0 such that for all 0 < x ≤ y ≤ r0,

(x

y

)2α+ε

≤ φ(x)φ(y)

≤(x

y

)2α−ε

. (3.11)

Consequently, we have

(i). limr→0φ(r)/r2 = ∞.

(ii). The function φ has the following doubling property: there exists a constantc3,3 > 0 such that φ(2r) ≤ c3,3 φ(r) for all 0 < r < r0/2.

17

Remark 3.3 Under the assumption that ∆ is absolutely continuous withdensity f(λ), Condition (3.10) is more general than assuming φ is regularlyvarying at 0. Using the terminology of Bingham et al. (1987, pp.65-67), (3.11)implies that φ is extended regularly varying at 0 with upper and lower Karamataindices 2α and 2α, respectively. A necessary and sufficient condition for φ(r) tobe regularly varying at 0 of index 2α is that the limit

α =12

limr→∞

rN∫

SN−1 f(rθ)µ(dθ)∆{ξ : |ξ| ≥ r}

exists; see Xiao (2005) for details. ¤

The following theorem shows that the assumption (3.10) implies that X isSLφND and φ(r) is comparable with σ2(h) with |h| = r near 0. In Section 4, wewill show that it is often more convenient to use the function φ to characterizethe properties of X.

Theorem 3.4 Let X = {X(t), t ∈ RN} be a mean zero, real-valued Gaussianrandom field with stationary increments and X(0) = 0. Assume that the spectralmeasure ∆ of X has a density function f that satisfies (3.10). Then

0 < lim infh→0

σ2(h)φ(|h|) ≤ lim sup

h→0

σ2(h)φ(|h|) < ∞. (3.12)

Moreover, for every interval T ∈ A, X is strongly locally φ-nondeterministic onT .

The first part of Theorem 3.4 is proved using the ideas of Berman (1988,1991) and the second part follows from (3.12) and Theorem 3.1; see Xiao (2005)for details.

Applying Theorems 3.1 and 3.4 to stationary Gaussian random fields, wehave the following partial extension of the result of Cuzick and DuPreez (1982)mentioned in section 2.2. It is not known to me whether (3.6) can be replacedby the weaker condition (2.11).

Corollary 3.5 Let X = {X(t), t ∈ RN} be a stationary Gaussian random fieldwith mean 0 and variance 1.

(i). If the spectral measure ∆ of X has an absolutely continuous part withdensity f satisfying (3.5) and (3.6), then for every interval T ∈ A, X isstrongly locally φ-nondeterministic on T .

(ii). If the spectral density of X satisfies (3.10), then (3.12) holds and X isSLφND on T .

We end this section with some more examples of Gaussian random fieldswhose SLND can be determined.

18

Example 3.6 Consider the mean zero Gaussian random field X = {X(t), t ∈RN} with stationary increments and spectral density

fγ,β(λ) =c(γ, β, N)

|λ|2γ(1 + |λ|2)β,

where γ and β are constants satisfying β + γ > N2 , 0 < γ < 1 + N

2 andc(γ, β, N) > 0 is a normalizing constant. Since the spectral density fγ,β involvesboth the Fourier transforms of the Riesz kernel and the Bessel kernel, Anh etal. (1999) call the corresponding Gaussian random field the fractional Riesz-Bessel motion with indices β and γ; and they have shown that these Gaussianrandom fields can be used for modelling simultaneously long range dependenceand intermittency.

It is easy to check that Condition (3.10) is satisfied with α = α = γ +β− N2 .

Moreover, since the spectral density fγ,β(x) is regularly varying at infinity oforder 2(β+γ) > N , by a result of Pitman (1968) we know that, if γ+β− N

2 < 1,then σ(h) is regularly varying at 0 of order γ + β −N/2 and

σ(h) ∼ |h|γ+β−N/2 as h → 0.

Theorem 3.4 implies that X is SLND with respect to σ2(h). Hence, manysample path properties of the d-dimensional fractional Riesz-Bessel motion Xwith indices β and γ can be derived from the results in Section 4.

Example 3.7 Let 0 < α < 1, 0 < c1 < c2 be constants such that (αc2)/c1 < 1.For any increasing sequence {bn, n ≥ 0} of real numbers such that b0 = 0 andbn →∞, define the function f on RN by

f(λ) ={

c1 |λ|−(2α+N) if |λ| ∈ (b2k, b2k+1],c2 |λ|−(2α+N) if |λ| ∈ (b2k+1, b2k+2].

(3.13)

Some elementary calculation shows that, when limn→∞ bn+1/bn = ∞, Condition(3.10) is satisfied with α = (αc1)/c2 < α = (αc2)/c1. Note that in this case,c1 |h|2α ≤ σ2(h) ≤ c2 |h|2α and c1 r2α ≤ φ(r) ≤ c2 r2α for all h ∈ RN and r > 0,but both functions are not regularly varying at the origin.

The following is a class of Gaussian random fields for which (3.10) does nothold, but Theorem 3.1 is still applicable.

Example 3.8 For any given constants 0 < α1 < α2 < 1 and any increasingsequence {bn, n ≥ 0} of real numbers such that b0 = 0 and bn →∞, define thefunction f on RN by

f(λ) ={ |λ|−(2α1+N) if |λ| ∈ (b2k, b2k+1],|λ|−(2α2+N) if |λ| ∈ (b2k+1, b2k+2].

(3.14)

Using such functions f as spectral densities, we obtain a quite large class ofGaussian random fields with stationary increments that are significantly differ-ent from the fractional Brownian motion. If X is such a random field, then

19

there exist positive and finite constants c3,4 , c3,5 ≥ 1 such that

c−13,4|h|2α2 ≤ σ2(h) ≤ c3,4 |h|2α1 , ∀ |h| ≤ 1 (3.15)

andc−13,5

r2α2 ≤ φ(r) ≤ c3,5 r2α1 , ∀ 0 < r ≤ 1. (3.16)

Xiao (2005) shows that we can choose the sequence {bn} appropriately so thatthe following hold:

(i) φ(r) ³ r2α2 for r ∈ (0, 1).

(ii) σ2(h) ³ |h|2α2 for |h| ≤ 1.

(iii) Condition (3.10) is not satisfied, but the corresponding Gaussian randomfield X still has the property of SLφND.

So far, we have not considered Gaussian random fields with stationary incre-ments and discrete spectral measures. A systematic treatment for such Gaussianrandom fields will be done elsewhere. In the following, we only give an exampleof stationary Gaussian processes with discrete spectrum that is strongly locallynondeterministic.

Example 3.9 Let {Xn, Yn, n ≥ 0} be a sequence of independent standardnormal random variables. Then for each t ∈ R, the random Fourier series

Y (t) =√

8π

∞∑n=0

12n− 1

{Xn cos

((2n− 1)t

)+ Yn sin

((2n− 1)t

)}(3.17)

converges almost surely [see, e.g., Kahane (1985)], and Y = {Y (t), t ∈ R} is acentered, periodic and stationary Gaussian process with mean 0 and covariancefunction

R(s, t) = 1− 2π|s− t| for − π ≤ s− t ≤ π. (3.18)

It is easy to see that the spectrum measure ∆ of Y is discrete with ∆({2n−1}) =(2n − 1)−2 for all n ∈ N. Using a result in Berman (1978), it can be shownthat for any interval T ⊂ [−π, π] with length |T | ≤ π/2 there exists a constant0 < c3,6 < ∞ such that for all t ∈ T and all 0 < r ≤ min{|t|, π/2},

Var(Y (t)

∣∣ Y (s) : s ∈ T, |s− t| ≥ r) ≥ c3,6 r. (3.19)

That is, Y is SLφND on T with φ(r) = r and σ2(h) ³ φ(|h|); see Shieh andXiao (2004) for a proof.

20

4 Applications of strong local nondeterminism

Many authors have applied LND or SLφND to study various properties ofGaussian and stable random fields. We refer to Geman and Horowitz (1980),Geman et al. (1984), Berman (1991), Dozzi (2002), and the references thereinfor more information.

In the studies of Gaussian random fields with stationary increments, thevariance function σ2(h) has played a significant role and it is typically assumedto be regularly varying at 0 and/or monotone increasing. See Marcus (1968b),Kono (1970, 1996), Cuzick (1982b), Csorgo et al. (1995), Kasahara et al. (1999),Monrad and Rootzen (1995), Talagrand (1995, 1998), Xiao (1996, 1997a, b,2003), and so on. Using the results in Section 3, we can prove that, in almost allcases, the regularly varying assumption on σ2(h) can be significantly weakenedand the monotonicity assumption can be removed.

In the rest of this section, we show that SLφND can be applied to extend thesmall ball probability estimates of Monrad and Rootzen (1995), Shao and Wang(1995) and Stoltz (1996), the results on the exact Hausdorff measure functionsof Talagrand (1995) and Xiao (1996, 1997a, b), the Holder conditions and tailprobability of the local times of Xiao (1997a) and Kasahara et al. (1999), tomore general Gaussian random fields. For proofs of these results, see Xiao(2005).

We will consider a Gaussian random field X = {X(t), t ∈ RN} in Rd definedby

X(t) =(X1(t), . . . , Xd(t)

), ∀t ∈ RN , (4.1)

where X1, . . . , Xd are independent copies of a real-valued, centered Gaussianrandom field Y = {Y (t), t ∈ RN}, which satisfies the following Condition (C):

(C1) there exist positive constants δ0, c4,1 , c4,2 and a non-decreasing, rightcontinuous function φ : [0, δ0) → [0,∞) such that

φ(2r)φ(r)

≤ c4,1 , ∀ r ∈ [0, δ0/2) (4.2)

and for all t ∈ RN and h ∈ RN with |h| ≤ δ0,

c−14,2

φ(|h|) ≤ E[(Y (t + h)− Y (t)

)2] ≤ c4,2φ(|h|). (4.3)

(C2) Y is strongly locally φ-nondeterministic on an interval T ∈ A, say, T =[0, 1]N .

4.1 Small ball probability and Chung’s law of the iteratedlogarithm

In recent years, there has been much interest in studying the small ball proba-bility of Gaussian processes. We refer to Li and Shao (2001) and Lifshits (1999)for extensive surveys on small ball probabilities, their applications and openproblems.

21

The next theorem gives estimates on the small ball probability of Gaussianrandom fields satisfying the condition (C). In particular, the upper bound in(4.4) confirms a conjecture of Shao and Wang (1995), under a much weakercondition.

Theorem 4.1 Let X = {X(t), t ∈ RN} be a Gaussian random field in R sat-isfying the condition (C). Then there exist positive constants c4,3 and c4,4 suchthat for all x ∈ (0, 1),

exp(− c4,3

[φ−1(x2)]N)≤ P

{max

t∈[0,1]N|X(t)| ≤ x

}≤ exp

(− c4,4

[φ−1(x2)]N), (4.4)

where φ−1(x) = inf{y : φ(y) > x} is the right-continuous inverse function of φ.

The lower bound in (4.4) follows from a general result of Talagrand (1993)and the upper bound is proved in a way similar to that of Monrad and Rootzen(1995). This is where SLφND of X is applied.

The probability estimate in Theorem 4.1 has many applications. We mentionthe following Chung’s law of the iterated logarithm. When σ is assumed to beregularly varying at 0, this is also obtained in Xiao (1997a).

Corollary 4.2 If, in addition to the conditions of Theorem 4.1, we assumethat X has stationary increments and the spectral measure ∆ of X satisfieslim infλ→∞ |λ|N+2∆

(B(λ, r)

)> 0, where B(λ, r) = {x ∈ RN : |x − λ| ≤ r}.

Then there exists a positive and finite constant c4,5 such that

lim infh→0

supt∈[0,h]N |X(t)|φ1/2

(h/(log log(1/h))1/N

) = c4,5 a.s. (4.5)

The proof of Corollary 4.2 contains two steps: first we apply Theorem 4.1and slightly modify the proof of Theorem 7.1 in Li and Shao (2001) to show theabove liminf is bounded from below and above by positive constants, then weapply the zero-one law of Pitt and Tran (1979) to derive (4.5).

We can also consider the small ball probability of Gaussian random fieldsunder the Holder-type norm. Let κ be a continuous and non-decreasing functionsuch that κ(r) > 0 for all r > 0. For any function y ∈ C0([0, 1]N ), we considerthe functional

‖y‖κ = sups,t∈[0,1]N ,s 6=t

|y(s)− y(t)|κ(|s− t|) . (4.6)

When κ(r) = rα, ‖ · ‖κ is the α-Holder norm on C0([0, 1]N ) and is denoted by‖ · ‖α.

The following theorem uses SLφND to improve the results of Stolz (1996).We mention that the conditions of Theorem 2.1 of Kuelbs, Li and Shao (1995)can be weakened in a similar way.

22

Theorem 4.3 Let X = {X(t), t ∈ RN} be a Gaussian random field in R satis-fying the condition (C). If for some constant β > 0,

φ1/2(r)κ(r)

³ rβ , ∀r ∈ (0, 1). (4.7)

Then there exist positive constants c4,6 and c4,7 such that for all ε ∈ (0, 1),

exp(− c4,6 ε−N/β

)≤ P

{‖X‖κ ≤ ε

}≤ exp

(− c4,7 ε−N/β

). (4.8)

4.2 Hausdorff dimension and Hausdorff measure of thesample paths

In this section we consider the fractal properties of the range and graph of theGaussian random field in Rd defined by (4.1) and fractional Brownian sheets.

When X = {X(t), t ∈ RN} is a fractional Brownian motion in Rd, the exactHausdorff measure functions for the image X([0, 1]N ) and graph GrX([0, 1]N )= {(t,X(t)) : t ∈ [0, 1]N} were determined by Talagrand (1995) and Xiao(1997c). Their results were extended by Xiao (1996, 1997a) to strongly locallynondeterministic Gaussian random fields with stationary increments and reg-ularly varying variance function σ2(h). Using the results in Section 3 we canprove the following more general result.

Theorem 4.4 Let X = {X(t), t ∈ RN} be a Gaussian random field defined in(4.1). We assume that Condition (C) is satisfied and, in addition, there existsa constant c4,8 > 0 such that

∫ ∞

1

( φ(a)φ(ax)

)d/2

xN−1dx ≤ c4,8 for all a ∈ (0, 1), (4.9)

then0 < ϕ1-m

(X([0, 1]N )

)< ∞ a.s., (4.10)

where ϕ1(r) =[φ−1(r2)

]N log log 1/r.

Remark 4.5 Note that condition (4.9) suggests that X is transient and doesnot hit points. When X hits points, the Hausdorff dimension of GrX([0, 1]N )may be bigger. The results on the exact Hausdorff measure of the graph setGrX([0, 1]N ) in Xiao (1997a, c) can be extended to Gaussian random fieldssatisfying Condition (C) in a similar way. ¤

Remark 4.6 More generally, Kahane (1985) has studied geometric and arith-metic properties of the image X(E) for an arbitrary Borel set E ⊂ RN when Xis an (N, d)-fractional Brownian motion. His results have recently be extendedand improved by Shieh and Xiao (2004). ¤

23

Now we give a brief discussion of the relevance of sectorial LND to the studyof fractal properties of fractional Brownian sheets. Further information can befound in Ayache, Wu and Xiao (2005), Khoshnevisan, Wu and Xiao (2005), Wuand Xiao (2005).

Recently, Ayache and Xiao (2004) have obtained the Hausdorff and packingdimensions of the range B

~H([0, 1]N ), graph GrB ~H([0, 1]N ) and the level set fora general (N, d)-fractional Brownian sheet B

~H . It would be interesting to findthe exact Hausdorff and packing measure functions for these random sets (ifthey exist). The existing methods for the Brownian sheet [cf. Ehm (1981)] orfractional Brownian motion [cf. Talagrand (1995), Xiao (1997c)] can not beapplied directly. I believe that the sectorial local nondeterminism of B

~H will beuseful in solving these problems.

Wu and Xiao (2005) have applied the sectorial local nondeterminism of B~H

to study geometric properties of the image set B~H(E), where E ⊂ (0,∞)N is

an arbitrary Borel set, of fractional Brownian sheets. In particular, they haveproved the following “uniform” Hausdorff and packing dimension result.

Theorem 4.7 Let H ∈ (0, 1) be a constant and let B~H = {B ~H(t), t ∈ RN

+}be an (N, d)-fractional Brownian sheet with index ~H = 〈H〉. If N ≤ Hd, thenalmost surely

dimHBH(E) =1H

dimHE for all Borel sets E ⊂ (0,∞)N . (4.11)

and

dimPBH(E) =1H

dimPE for all Borel sets E ⊂ (0,∞)N . (4.12)

Note that when ~H = 〈 12 〉, (4.11) of Theorem 4.7 recovers the result for the(N, d)-Brownian sheet W = {W (t), t ∈ RN

+} proved by Mountford (1989) andLin (1999); see also Khoshnevisan, Wu and Xiao (2005) for a different, relativelymore elementary proof. Both the proofs of Mountford (1989) and Lin (1999)are quite involved and their arguments rely on the special properties of theBrownian sheet such as the independence of the increments, which can not beapplied to fractional Brownian sheets. Our proof of Theorem 4.7 is, similarto that in Khoshnevisan, Wu and Xiao (2005), based on the sectorial localnondeterminism of fractional Brownian sheets.

Finally we mention that when ~H = (H1, . . . , HN ) ∈ (0, 1)N and H1, . . . , HN

are different, the Hausdorff and packing dimension of E alone is not enoughfor determining dimHB

~H(E) or dimPB~H(E). Explicit formulas for dimHB

~H(E)and dimPB

~H(E) are not known.

4.3 Local times and level sets of Gaussian random fields

Let X = {X(t), t ∈ RN} be a Gaussian random field with stationary incrementsin Rd defined by (4.1). If the real-valued random field Y satisfies Condition (C)

24

and for some ε > 0, ∫

[0,1]N

dh

σd+ε(h)< ∞, (4.13)

then it follows from Theorem 26.1 in Geman and Horowitz (1980) [see alsoBerman (1973) and Pitt (1978)] that X has a jointly continuous local timeL(x, t) for (x, t) ∈ Rd × T which satisfies certain Holder conditions in the timeand space variables, respectively.

When X is strongly locally nondeterministic and satisfies certain regularityassumptions, Xiao (1997a) has established sharp local and uniform Holder con-ditions for the local time L(x, t) in the time variable t. Besides of their owninterest, these Holder conditions are also useful in studying the fractal prop-erties of the sample paths of X. In the following, we show that the results inXiao (1997a) and Kasahara et al. (1999) still hold under Condition (C). Forsimplicity, we will only consider the case N = 1.

Theorem 4.8 Let X = {X(t), t ∈ R} be a centered Gaussian process in Rd

defined by (4.1). We assume that the associated Gaussian process Y satisfiesCondition (C) and there exist constants 0 < γ0 < 1 and c4,9 > 0 such that

∫ 1

0

( φ(a)φ(as)

) d2 +γ0

ds ≤ c4,9 for all a ∈ (0, δ0). (4.14)

For any B ∈ B(R) define L∗(B) = supx∈Rd L(x, B). Then there exists a positiveand finite constant c4,10 such that for all t ∈ R,

lim supr→0

L∗(B(t, r))ϕ2(r)

≤ c4,10 a.s. (4.15)

and for all intervals T ⊆ R, there exists a positive and finite constant c4,11 suchthat

lim supr→0

supt∈T

L∗(B(t, r))ϕ3(r)

≤ c4,11 a.s., (4.16)

where B(t, r) = (t− r, t + r),

ϕ2(r) =r

φ(r(log log 1/r)−1)d/2and ϕ3(r) =

r

φ(r(log 1/r)−1)d/2.

Remark 4.9 If X has stationary increments and its spectral measure satisfies(3.10) and 1 > α d. Then Lemma 3.2 implies that (4.14) is satisfied for anyγ0 ∈ (0, (1− αd)/(2α)). ¤

Similar to Xiao (1997a), the proof of Theorem 4.8 is based on the momentestimates for L(x,B) and L(x + y, B)−L(x,B) and a chaining argument. Thefollowing lemma provides the key estimates, whose proofs rely on SLφND of X.

25

Lemma 4.10 Under the conditions of Theorem 4.8, there exist positive andfinite constants c4,12 and c4,13 such that for all integers n ≥ 1, r > 0 small,x ∈ Rd and 0 < γ < γ0 , we have

E[L(x, r)n

] ≤ cn4,12

rn

φ(r/n)nd/2(4.17)

and

E[L(x + y, r)− L(x, r)

]n ≤ cn4,13|y|nγ rn

[σ(r/n)

](d+2γ)n/2(n!)γ . (4.18)

Theorem 4.8 can be applied to determine the Hausdorff dimension and ex-act Hausdorff measure of the level set X−1(x) = {t ∈ R : X(t) = x}, wherex ∈ Rd. For example, the Hausdorff dimension of X−1(x) has been studiedby Berman (1970, 1972), Adler (1981), Monrad and Pitt (1987) for index α-Gaussian processes; and the exact Hausdorff measure of X−1(x) has been stud-ied by Xiao (1997a) for a class of strongly locally nondeterministic Gaussianrandom fields with stationary increments.

By applying (4.16) of Theorem 4.8, Xiao (2005) has proved the followinguniform Hausdorff dimension result for the level sets of the Gaussian process X,extending the previous results of Berman (1972), Monrad and Pitt (1987).

Theorem 4.11 Let X = {X(t), t ∈ R} be a Gaussian process in Rd with sta-tionary increments defined by (4.1). We further assume that Y satisfies theassumptions of Theorem 3.4. Then with probability one,

dimHX−1(x) = 1− α∗d for all x ∈ O, (4.19)

where α∗ is the upper index of σ defined by

α∗ = inf{

γ ≥ 0 : limh→0

σ(h)|h|γ = ∞

}

with the convention inf ∅ = ∞, and where O is the (random) open set definedby

O =⋃

s,t∈Q; s<t

{x ∈ Rd : L(x, [s, t]) > 0

}.

.

Remark 4.12 It is an interesting question to characterize the random openset O. Monrad and Pitt (1987) have given a real-valued periodic stationaryGaussian process X for which O is a proper subset of R [because the range of Xis a.s. bounded]. They have shown a sufficient condition in terms of the spectralmeasure of a stationary (N, d)-Gaussian random field X so that O = Rd holds.Monrad and Pitt (1987) also point out that the self-similarity of an (N, d)-fractional Brownian motion Bα implies that if N > αd then O = Rd almostsurely. However, we do not know whetherO = Rd is true for the (N, d)-Gaussianrandom fields satisfying the conditions of Theorem 3.4. ¤

26

The local time L(0, 1) [i.e., L(x, 1) at x = 0] of a Gaussian process X some-times appears as limit in some limit theorems on the occupation measure of X;see, for example, Kono (1996) and Kasahara and Ogawa (1999). Since there islittle knowledge on the explicit distribution of L(0, 1), it is of interest in esti-mating the tail probability P{L(0, 1) > x} as x → ∞. This problem has beenconsidered by Kasahara et al. (1999) under some quite restrictive conditions onthe Gaussian process X. The next theorem is an extension of the main resultin Kasahara et al. (1999).

Theorem 4.13 Let X = {X(t), t ∈ R} be a centered Gaussian process in Rd

defined by (4.1). We assume that the associated process Y satisfies Condition(C) and the condition (4.14) with γ0 = 0. Then

− logP{L(0, 1) > x

} ³ 1φ−1(1/x2)

, (4.20)

where φ−1 is the inverse function of φ as defined in Theorem 4.1.

Theorem 4.13 follows from the moment estimates for L(0, 1) in Lemma 4.15and the following lemma on the tail probability of nonnegative random variables.When ψ is a power function or a regularly varying function, Lemma 4.14 is wellknown.

Lemma 4.14 Let ξ be a non-negative random variable and let ψ : R+ → R+ bea non-decreasing function having the doubling property. If there exist positiveconstants c4,14 and c4,15 such that

cn4,14

ψ(n)n ≤ E(ξn) ≤ cn4,15

ψ(n)n

for all n large enough, then there exist positive constants c4,16 > c4,15 , c4,17 andc4,18 such that for all x > 0 large enough,

e−c4,17x ≤ P{ξ ≥ c4,16 ψ(x)

} ≤ e−c4,18x. (4.21)

Lemma 4.15 Under the assumptions of Theorem 4.13, there exist positive andfinite constants c4,19 and c4,20 such that for all integers n ≥ 1,

cn4,19

φ(1/n)nd/2≤ E[

L(0, 1)n] ≤ cn

4,20

φ(1/n)nd/2. (4.22)

One may also consider the existence and continuity of the local times L(x,E)of an (N, d)-Gaussian random field on any Borel set E ⊂ RN . These prob-lems are closely related to the questions whether the image X(E) has positiveLebesgue measure and/or interior points; see Pitt (1978), Kahane (1985), Shiehand Xiao (2004), Khoshnevisan and Xiao (2004b). Strong local nondetermin-ism and sectorial local nondeterminism have proven to be very useful for solvingthese problems. On the other hand, similar to Theorem 4.13, the distributionof L(0, E) can be studied for a large class of fractal sets, say d-sets.

We end this section with the following remarks and open questions.

27

Remark 4.16 The problem of establishing sharp uniform and local Holderconditions for the self-intersections local times of Gaussian random fields sat-isfying Condition (C) remains to be open. For background and some relatedresults, see Berman (1991). ¤

Remark 4.17 Using sectorial local nondeterminism, Ayache, Wu and Xiao(2005) have established joint continuity and sharp Holder conditions for thelocal times of a fractional Brownian sheet B

~H . Their results suggest someinteresting questions for general anisotropic Gaussian random fields that maybe further investigated. ¤

Question 4.18 We know that (4.13) is sufficient for the existence of a jointlycontinuous local time of locally nondeterministic Gaussian random field X.However, this condition is not necessary. When X is an (N, d)-Gaussian ran-dom field with stationary increments, is it possible to provide a necessary andsufficient condition for the joint continuity of L(x, t) in terms of φ?

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Yimin Xiao. Department of Statistics and Probability, A-413 Wells Hall,Michigan State University, East Lansing, MI 48824E-mail: xiao@stt.msu.edu

URL: http://www.stt.msu.edu/~xiaoyimi

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